3.9.14 \(\int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac {2 a}{5 x^{5/2}}-\frac {2 b}{\sqrt {x}}+\frac {2}{3} c x^{3/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \begin {gather*} -\frac {2 a}{5 x^{5/2}}-\frac {2 b}{\sqrt {x}}+\frac {2}{3} c x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x] + (2*c*x^(3/2))/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx &=\int \left (\frac {a}{x^{7/2}}+\frac {b}{x^{3/2}}+c \sqrt {x}\right ) \, dx\\ &=-\frac {2 a}{5 x^{5/2}}-\frac {2 b}{\sqrt {x}}+\frac {2}{3} c x^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 a-15 b x^2+5 c x^4\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(2*(-3*a - 15*b*x^2 + 5*c*x^4))/(15*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 a-15 b x^2+5 c x^4\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(2*(-3*a - 15*b*x^2 + 5*c*x^4))/(15*x^(5/2))

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fricas [A]  time = 1.75, size = 21, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (5 \, c x^{4} - 15 \, b x^{2} - 3 \, a\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c*x^4 - 15*b*x^2 - 3*a)/x^(5/2)

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giac [A]  time = 0.21, size = 20, normalized size = 0.69 \begin {gather*} \frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="giac")

[Out]

2/3*c*x^(3/2) - 2/5*(5*b*x^2 + a)/x^(5/2)

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maple [A]  time = 0.00, size = 22, normalized size = 0.76 \begin {gather*} -\frac {2 \left (-5 c \,x^{4}+15 b \,x^{2}+3 a \right )}{15 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^(7/2),x)

[Out]

-2/15*(-5*c*x^4+15*b*x^2+3*a)/x^(5/2)

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maxima [A]  time = 1.08, size = 20, normalized size = 0.69 \begin {gather*} \frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="maxima")

[Out]

2/3*c*x^(3/2) - 2/5*(5*b*x^2 + a)/x^(5/2)

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mupad [B]  time = 4.33, size = 21, normalized size = 0.72 \begin {gather*} -\frac {-10\,c\,x^4+30\,b\,x^2+6\,a}{15\,x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^(7/2),x)

[Out]

-(6*a + 30*b*x^2 - 10*c*x^4)/(15*x^(5/2))

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sympy [A]  time = 1.87, size = 27, normalized size = 0.93 \begin {gather*} - \frac {2 a}{5 x^{\frac {5}{2}}} - \frac {2 b}{\sqrt {x}} + \frac {2 c x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**(7/2),x)

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x) + 2*c*x**(3/2)/3

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